Question: Simplify and expand the following expression: $ \dfrac{k}{k + 3}+\dfrac{k}{5k + 1} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(k + 3)(5k + 1)$ Multiply the first term by $\dfrac{5k + 1}{5k + 1}$ $ \begin{align*} \dfrac{k}{k + 3} \times \dfrac{5k + 1}{5k + 1} & = \dfrac{(k)(5k + 1)}{(k + 3)(5k + 1)} \\ & = \dfrac{5k^2 + k}{(k + 3)(5k + 1)}\end{align*} $ Multiply the second term by $\dfrac{k + 3}{k + 3}$ $ \begin{align*} \dfrac{k}{5k + 1} \times \dfrac{k + 3}{k + 3} & = \dfrac{(k)(k + 3)}{(5k + 1)(k + 3)} \\ & = \dfrac{k^2 + 3k}{(5k + 1)(k + 3)}\end{align*} $ Now we have: $ = \dfrac{5k^2 + k}{(k + 3)(5k + 1)} + \dfrac{k^2 + 3k}{(5k + 1)(k + 3)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{5k^2 + k + k^2 + 3k}{(k + 3)(5k + 1)} $ $ = \dfrac{6k^2 + 4k}{(k + 3)(5k + 1)}$ Expand the denominator: $ = \dfrac{6k^2 + 4k}{5k^2 + 16k + 3}$